There are many different types of function tables. Here, you’ll look at two types: One that is nice in the beginning when you are learning, because it contains a lot of detail, and one that is a bit more concise and professional.
A function table is a table that shows which coordinates should be plotted in the coordinate system, so that you can draw the graph of the function.
A comprehensive function table has four columns: A column for the $x$-values you need, a column for the calculation of each $y$-value, a column where the $y$-values that were calculated is written, and a column for the coordinates you get from the chosen $x$-value and the corresponding $y$-value so that you can plot the graph. It looks like this:
In the first column, you place the $x$-values that you want to examine. Why did I choose exactly these values? Well, which values you need follow from the problem you’re working on. Plus, you are often told which $x$-values you should examine. If you can’t figure it out, you can start with values between $-5$ and 5. This is a good place to begin.
In the second column, you see the function $y=2x+3$. This is a function of a straight line. In this column, you insert the chosen $x$-values into the function and calculate the $y$-values.
Then, you write the corresponding $y$-values in the third column. If you calculate the contents in every cell in the second column, you see that the answers are exactly the values that are written in the third column.
Finally, in the fourth column, the $x$-value and $y$-value are put together into coordinates $(x,y)$. These are the coordinates you have to insert in the coordinate system and draw lines through to make your graph.
The professional function table contains precisely what you need to find the coordinates, so that you can draw the graph. The difference from the previous function table is that Columns 2 and 4 are removed, and you write the table horizontally instead of vertically (changing columns into rows).
Even if you don’t include the coordinate column, you see that the $x$-values and $y$-values are included, and it is just these that make up the coordinates. In other words, $(x,y)$ is included, it just looks a little different.
Here’s the graph with the points marked:
Example 1
Lance is $\text{}30\text{}\phantom{\rule{0.17em}{0ex}}\text{km}$ from home. He rides his bike at a constant speed of $\text{}12\text{}\phantom{\rule{0.17em}{0ex}}\text{km/h}$. Make a graphical illustration of the number of hours and the number of kilometers he is from home. How much time does it get before he gets home?
To solve this problem, you should first convert the information from the exercise into a function. The formula for a straight line is most suitable, since Lance is riding his bike at a constant speed.
Let’s examine how words and expressions are converted to mathematical symbols. When solving this problem you should think like this: Text informs us that Lance is $30$ km away from home and that he is riding his bike at $12$ km/h. So his journey begins at $30$ km distance, and the closer he gets to home, the shorter the distance. Thus, you convert the information like this:
$f(x)$ is the distance to home at any given time. This distance gets smaller and smaller, the longer Lance travels towards home.
$x$ is the number of hours Lance rides his bike.
$30$ km is the distance that is counted down to $0$ km, from where Lance starts, until he reaches home.
$12$ km/h is Lance’s speed.
Now you insert these items into the formula for a straight line, $f(x)=mx+b$, and get:
$$f(x)=-12x+30$$ |
30 is the constant term since the starting point for Lance is set—$30$ km from home. $-12$ is the slope, since the distance to home is reduced by $12$ km for every hour Lance rides his bike towards home.
Note! The minus sign for Lance’s speed does not mean that he’s riding his bike in reverse all the way home! Instead, the minus sign is used to count down from $30$ km to $0$ km. The distance should decrease, so you need a minus sign to accomplish this.
Based on the expression above you get the following function table:
$x$ | $y=f(x)=-12x+30$ | $(x,y)$ | |
$0$ | $f(0)=-12\cdot 0+30=30$ | $(0,30)$ | |
$0.5$ | $f(0.5)=-12\cdot 0.5+30=24$ | $(0.5,24)$ | |
$1$ | $f(1)=-12\cdot 1+30=18$ | $(1,18)$ | |
$1.5$ | $f(1.5)=-12\cdot 1.5+30=12$ | $(1.5,12)$ | |
$2$ | $f(2)=-12\cdot 2+30=6$ | $(2,6)$ | |
$2.5$ | $f(2.5)=-12\cdot 2.5+30=0$ | $(2.5,0)$ | |
$3$ | $f(3)=-12\cdot 3+30=-6$ | $(3,-6)$ | |
$$ |
From the function table, you see that Lance gets home after $x=2.5$ hours. Since the problem asks you to solve this graphically, you now have to make a coordinate system in which you mark the points you have calculated and written in the column with $(x,y)$, and then draw the line between them. The graph should look like this:
Lance arrives at home when the $y$-coordinate is 0. This is when the graph intersects the $x$-axis. The circle marks this point and you can see that Lance arrives at home after $x=2.5$ hours. Thus, the answer is:
It takes $2.5$ hours for Lance to arrive at home.